Product Description
A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: is there a "best" bend and, if so, what is it?" Most of the well-known bends happen to be symmetric - that is, the two constituent cords within the bend have the same geometric shape and size, as well as an interrelationship. Such "symmetric bends" have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage of being easier to tie (with less chance of error), and of probably being stronger, since neither end is the weaker. This book presents a mathematical theory of symmetric bends, together with a simple explanation of how such bends may be invented. Also discussed are the additionally symmetric "triply symmetric" bends. Full details, including colour pictures, are given of the "best 60" known symmetric bends, many of which were created by those methods of invention. This work will appeal to many - mathematicians as well as non-mathematicians interested in beautiful and useful knots. ... Read more

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This volume includes both asymptotic results on the inevitability of random knotting and linking, and Monte Carlo simulations of knot probability at small lengths. The statistical mechanics and topology of surfaces on the d-dimensional simple cubic lattice are investigated. The energy of knots is studied both analytically and numerically. Vassiliev invariants are investigated and used in random knot simulations. A mutation scheme which leaves the Jones polynomial unaltered is described. Applications include the investigation of RNA secondary structure using Vassiliev invariants, and the direct experimental measurement of DNA knot probability as a function of salt concentration in random cyclization experiments on linear DNA molecules. The papers in this volume reflect the diversity of interest across science and mathematics in this subject, from topology and statistical mechanics to theoretical chemistry and wet-lab molecular biology. ... Read more

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Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a "canonical configuration" of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. ... Read more

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One of the most significant unsolved problems in mathematics is the complete classification of knots. The main purpose of this text is to introduce the reader to the use of computer programming to obtain the table of knots. The author seeks to present this problem as clearly and methodically as possible, starting from the very basics. Mathematical ideas and concepts are extensively discussed, and no advance background is required. ... Read more

Customer Reviews (1)

An extraordinary book!
This book describes how to tabulate knots with the help of computers. You could ask, so, what is special about that and why is this difficult? It is difficult because a knot or link is a topological object and hence, it is not clear how to represent such an object that it can be processed by an computer at all. Even if you know the Dowker-Thistlethwaite notation it is still an art to do it.
In my opinion this book is ahead of its time, because the field of computational topology/knot theory is still in its infacncy. ... Read more

Product Description
This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik–Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants. ... Read more

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This monograph presents a complete and self-contained solution to the long-standing problem of giving a geometric description of state spaces of C*-algebras and von Neumann algebras, and of their Jordan algebraic analogs (JB-algebras and JBW-algebras). The material, which previously has appeared only in research papers and required substantial prerequisites for a reader's understanding, is made accessible here to a broad mathematical audience.

Key features include:

* The properties used to describe state spaces are primarily of a geometric nature, but many can also be interpreted in terms of physics. There are numerous remarks discussing these connections

* A quick introduction to Jordan algebras is given; no previous knowledge is assumed and all necessary background on the subject is given

* A discussion of dynamical correspondences, which tie together Lie and Jordan structures, and relate the observables and the generators of time evolution in physics

* The connection with Connes' notions of orientation and homogeneity in cones is explained

* Chapters conclude with notes placing the material in historical context

* Prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis

* Excellent bibliography and index

In the authors' previous book, "State Spaces of Operator Algebras: Basic Theory, Orientations and C*-products" (ISBN 0-8176-3890-3), the role of orientations was examined and all the prerequisites on C*-algebras and von Neumann algebras, needed for this work, were provided in detail. These requisites, as well as all relevant definitions and results with reference back to "State Spaces," are summarized in an appendix, further emphasizing the self-contained nature of this work.

"Geometry of State Spaces of Operator Algebras" is intended for specialists in operator algebras, as well as graduate students and mathematicians in other areas. ... Read more

Product Description

This monograph is Part 1 of a book project intended to give a full account of Jorgensen's theory of punctured torus Kleinian groups and its generalization, with application to knot theory.

Although Jorgensen's original work was not published in complete form, it has been a source of inspiration. In particular, it has motivated and guided Thurston's revolutionary study of low-dimensional geometric topology.

In this monograph, we give an elementary and self-contained description of Jorgensen's theory with a complete proof. Through various informative illustrations, readers are naturally led to an intuitive, synthetic grasp of the theory, which clarifies how a very simple fuchsian group evolves into complicated Kleinian groups.

Product Description
The aim of these notes is to link algorithmic problems arising in knot theory with statistical physics and classical combinatorics. Apart from the theory of computational complexity needed to deal with enumeration problems, introductions are given to several of the topics, such as combinatorial knot theory, randomized approximation models, percolation, and random cluster models. ... Read more

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Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. The book begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories, as well as lattice techniques.The authors then discuss applications in quantum gravity in detail.Following chapters consider knot theories, braid theories and extended loop representations in quantum gravity.A final chapter assesses the current status of the theory and points out possible directions for future research. ... Read more

Customer Reviews (1)

Good introduction to some of the early research
The search for a theory of quantum gravity has occupied the time of a large number of researchers for over half a century, but as yet there is hardly any agreement on the conceptual foundations of such a theory, and this situation is aggravated by the lack of any experimental results that would drive its construction. Such evidence is absolutely necessary, for the lack of it sometimes leads the researcher into making wild speculations, which even though they are interesting from a mathematical standpoint, cannot be distinguished from alternatives that also have no experimental foundation. This situation has not prevented researchers from trying to find a theory of quantum gravity, and some of them proceed by analogy to what is done in quantum field theories that are well understood, such as quantum electrodynamics. Other researchers, particularly those that have a sophisticated mathematical background, have chosen string theory as the best candidate for a quantum theory of gravity.

As is readily apparent in the forward to this book, the authors favor the first approach, believing that quantum gauge theories, of which quantum electrodynamics is a primary example, offer the best hope for guidance in constructing a viable quantum gravity. They emphasize though a very particular aspect of these theories, namely that the requirement for gauge invariance forces one to view the "Wilson loops" as being the entities of primary importance. But more importantly, the authors assert that that Wilson loops allow one to gain information in the non-perturbative realm of quantum field theory. Calculations in non-perturbative quantum field theory are notoriously difficult, even though some progress has been made in the area of lattice gauge theories, so any insight the authors can offer in this regard is of utmost importance. Hence this book should be viewed as a study of quantum observables on the loop space. The authors hope that these observables, called `Schwinger functions' in the perturbative realm, will along with the differential equations and boundary conditions that determine them, will give a viable theory of quantum gravity.

The differential geometry of gauge theories is usually done using the formalism of principal fiber bundles. Classical gauge fields are viewed as sections of these bundles, and the results of non-trivial field interactions are compared from point to point by the use of parallel transport along curves defined in the base spaces of these bundles. This comparison is done with a `connection' on the bundle, and for a closed curve the failure of an entity to return to its original value after traversing the curve is taken to be a sign of non-trivial interactions or "curvature". Principal fiber bundles of course have an associated Lie group and elements of this group act on objects to parallel transport them along the closed curves. These group elements are thus dependent on the curve, and are called `holonomies'.

This is the classical picture, but what happens to this scenario in the quantum realm, and in this realm is it plausible to view it as a theory of quantum gravity?The authors spend the first six chapters discussing the loop group, and its use in the quantization of classical electrodynamics and classical Yang-Mills theory, as compared with what is done in the usual Hamiltonian formalism. The `quantum loop representation' plays a central role in their exposition, which is motivated by essentially two different approaches, one of which is essentially a Fourier transform of wavefunctions of the connection, while the other involves the quantization of a non-canonical algebra. In both cases the quantization procedure involves coming to grips with a constrained system, which as is well known is very challenging and the loop representation cannot be expected to be a panacea in this regard.

The trick involves the identification of the physical states taking into account diffeomorphism invariance and the Hamiltonian constraint. The authors do this for pure quantum gravity (no matter fields) using the Ashtekar formalism and `point-splitting' methods, reinforcing the idea of course that one is not going to escape the need for regularization, as is the case for all successful quantum field theories so far. The inclusion of matter fields is done for (uncharged) Weyl fermions, with the geometric interpretation that the Weyl part of the Hamiltonian is a translation operator in much the same way as the Hamiltonian in the case of pure gravity. The authors believe take this to mean that the loop representation for quantum gravity predicts the Dirac equation for fermions, but unfortunately they do not elaborate on this in much detail at all.

If loop quantization is to bring about a "unified" field theory in some sense then it must be able to show how the loops from one theory can be combined with the loops from another. The authors do this for the case of general relativity and electrodynamics, wherein a loop representation is introduced that is based on a single loop that accounts for the information of these two interacting theories. As expected, this involves enlarging the symmetry group SU(2) to U(2). They show that the wavefunctions for the unified loop representation depend on two loops, but that there is no effective distinction between the two loops. Generalizations to the case of Yang-Mills + general relativity are alluded to in the text but not discussed in any depth.


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Product Description
This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose.

The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds. ... Read more

Product Description
In this book, the author announces the class of problemscalled "entropy of knots" and gives an overview of modern physicalapplications of existing topological invariants.

He constructs statistical models on knot diagrams and braids using therepresentations of Jones-Kauffman and Alexander invariants and putsforward the question of limit distribution of these invariants forrandomly generated knots. The relation of powers of correspondingalgebraic invariants to the Lyapunov exponents of the products ofnoncommutative matrices is described. Also the problem of conditionaljoint limit distributions for "brownian bridges" on braids isdiscussed. Special cases of noncommutative groups PSL(2,R), PSL(2,Z)and braid groups are considered in detail.

In this volume, the author also discusses the application of conformalmethods for explicit construction of topological invariants for randomwalks on multiconnected manifolds. The construction of thesetopological invariants and the monodromy properties of correlationfunction of some conformal theories are also discussed.The authoralso considers the physical applications of "knot entropy" problem invarious physical systems, focussing on polymers. ... Read more

Product Description
This book is the result of an AMS Short Course on Knots and Physics that was held in San Francisco (January 1994). The range of the course went beyond knots to the study of invariants of low dimensional manifolds and extensions of this work to four manifolds and to higher dimensions. The authors use ideas and methods of mathematical physics to extract topological information about knots and manifolds.

Features:

A basic introduction to knot polynomials in relation to statistical link invariants.

Concise introductions to topological quantum field theories and to the role of knot theory in quantum gravity.

Knots and Physics would be an excellent supplement to a course on algebraic topology or a physics course on field theory. ... Read more

Product Description
To attack certain problems in 4-dimensional knot theory the author draws on a variety of techniques, focusing on knots in S^T4, whose fundamental groups contain abelian normal subgroups. Their class contains the most geometrically appealing and best understood examples. Moreover, it is possible to apply recent work in algebraic methods to these problems. New work in four-dimensional topology is applied in later chapters to the problem of classifying 2-knots. ... Read more

Product Description
This journey through the fascinating world of molecular topology focuses on catenanes, rotaxanes and knots, their synthesis, properties, and applications and the theory of interlocking and interpenetrating molecules. Nearly one hundred years of progress have passed since Willstatter's speculative vision of a molecule consisting of two interlinked rings. But even today the synthesis of such structures are a challenge to the creativity of synthetic chemists. These molecules are not only of academic interest, since they occur naturally. In such molecules as DNA, knots and related topological features play a key role in biochemical processes. In addition, extensive research on the properties of polyrotaxanes and polycatenanes show potential applications as molecular magnets, wires or switches.

Twelve international leading experts in the field present the broad and impressive spectrum of the topology of these molecules, from theoretical aspects and new pathways in synthesis to probing their properties. All researchers working in this interdisciplinary area, whether organic, inorganic or polymer chemists, as well as material scientists, will welcome this comprehensive and up-to-date work as an inspiring source for creative research ideas.
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Product Description
This book presents a collection of papers on two related topics: topology of knots and knot-like objects (such as curves on surfaces) and topology of Legendrian knots and links in 3-dimensional contact manifolds.

Featured is the work of international experts in knot theory ("quantum" knot invariants, knot invariants of finite type), in symplectic and contact topology, and in singularity theory. The interplay of diverse methods from these fields makes this volume unique in the study of Legendrian knots and knot-like objects such as wave fronts. A particularly enticing feature of the volume is its international significance. The volume successfully embodies a fine collaborative effort by worldwide experts from Belgium, France, Germany, Israel, Japan, Poland, Russia, Sweden, the U.K., and the U.S. ... Read more

Customer Reviews (1)

Professionally done, top in field
An impressive collection on specific elaboration of topological concepts.The author's teaching experience shows throughout the book in its clear, elegant explanations of theory.Proofs are optimally outlined withappropriate illustration.I've recommended the book to my graduatestudents for review, and have seen productive theses in which the work hasbeen credited as a major source. ... Read more

Product Description
In geometric knot theory, a central issue is to study the various geometric properties of knots when the knots have certain thickness. This setting makes a knot more like one that is tied with a uniform physical rope. These problems are mostly motivated by the recent applications of knot theory in fields such as biology and polymer chemistry. In this book, the authors first give a brief review of the basic concepts and terminologies such as the thickness of a knot and the rope length of a knot. They then review the main results in this field. The topics include results on the global minimum rope length of knots, various lower and upper rope length bounds of knots in terms of their crossing numbers, and lower and upper bounds on the total curvatures of thick knots. Some special families of knots or under different settings, such as lattice knots and smooth knots are also considered. While some proofs are omitted or only outlined due to the page limitation of the book, many important ideas, methods, and theorems are explained in-depth. At the end of the book, a list of some open problems in this field is given. ... Read more